This paper discussed problems associated with numerical simulation of transient processesin n a natural gasp ipeline. This discussion includes such problems as the properties of existing numerical methods for solving differential equations, problems associated with applying these methods for simulating a natural gas pipeline, the issue of computational complexity associated with such simulation, as well as the applicability of a new class of numerical methods for solving differential equations, namely the explicit methods with extended stability region. One particular family of these methods discussed in the paper is known as the Runge-Kutta-Chebyshev family. The first part of the paper has a tutorial character. The topics discussed in this part include the properties of standard explicit numerical methods for solving ordinary differential equations, the sources of problems associated with selection of discretization step, and the principles of constructing explicit methods with extended stability region. The dynamical properties of a pipeline segment are also discussed in this part of the paper. The second part of the paper presents the Runge-Kutta-Chebyshev method, discusses its properties and implementation issues, and presents the preliminary results of applying this method for solving the equation of a pipeline.

1 Introduction

It is well known that numerical simulation of natural gas pipeline systems is associated with high computational cost. This high cost is caused by the nature of physical processes taking place in a pipeline and consequently, the dynamical properties of equations describing these processes. The process of gas flow consists of two phenomena: slow movement of mass, and fast transfer of energy associated with propagating the sound waves. Consequently, the solution of equations of a pipeline contains both very fast and very slow components. Due to the properties of numerical methods used for solving equations of a pipeline, the required time discretization step is determined by the fastest components of the solution and is usually extremely small. However, when studying the behavior of the mass flow processes is a main purpose of the simulation experiment, these fast components are negligible.

To overcome the difficulties associated with the selection of discretization step, the implicit methods are commonly used for solving pipeline equations. These methods require that a set of nonlinear algebraic equations must be solved for every time transition. Computationally, solving these nonlinear equations is equivalent to a steady state simulation of a complete network. Partial differential equations describing the flow of gas in a pipeline are well known and are discussed in several publications (Hannah at all., 1964, Lewandowski and Pacut, 1978, Wylie and Streeter, 1985, Osiadacz, 1987). After several simplifications, the mathematical model of a pipeline is formulated as a set of nonlinear partial differential equations of hyperbolic type. The majority of known numerical methods for solving the equations of a pipeline are based on the discretization in time and the discretization in space: only the values of gas pressure and gas flow in selected points of a pipeline segment are calculated, and these values are obtained only for selected time instants.

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